1 is a generator for the set of integers?

Question

I'm reading a text here that explains that the members of a cyclic group would be:

$x^{-2}, x^{-1}, x^0, x^1, x^2, x^3,...$

It goes on to say that,

The basic element x is considered to be the generator of the group, since all other members of the group are derived from it. It is also referred to as a primitive element of the group.

Here's the part that sounds wrong to me:

Integers could be considered a cyclic group, with 1 being the primitive element (the generator). All integers can be expressed as a power of 1 in this group.

Would this not be saying that all integers can be represented with the form $1^x$ where x is a member of the set of integers?

Answer 1

If $a*b = a+b $ then $a^3=a*a*a=a+a+a=3\times a $ and therefore, yes, it would be saying $\mathbb Z =\{1^x|x\in\mathbb Z\} =\{x\times 1|x\in \mathbb Z\} $. It would be saying that and it would be correct.

But that would be inconsistent notation. We usually use $*, a^n, 1=identity $ for groups even though $*$ doesn't need to mean "multiplication" and 1 doesn't need to be the number 1. To avoid confusion we sometimes use $+,n\circ a,0=identity $. This doesn't always mean addition either be we tend to use it when we do mean "addition".

But yes $( \mathbb Z, +)= <1,+> = \{n\circ 1\} $ is how we'd usually notation the statement that the integers is a cyclic group under addition generated by $1$.

Answer 2

The operation in that group is addition. So it should be "all integers are of the form $\underbrace{1+\cdots+1}_x$."

Answer 3

A generator in a cyclic group is the element which generates the other elements of the group. In your case, integers under addition, that means that the element 1 generates the rest of the elements of the group.

It also satisfies some properties: existence of inverse, identity element, commutativity, closure under addition and associativity.

When you are working with multiplicative integers mod p then the definition of a generator is the same: an element that generates the whole group, its order being $\phi(n)$. Then it receives the name primitive-root mod n, but generator is still valid.